Existence and Uniqueness of Positive Solutions to Three Coupled Nonlinear Schrdinger Equations

In this paper, we consider existence and uniqueness of positive solutions to three coupled nonlinear Schrdinger equations which appear in nonlinear optics. We use the behaviors of minimizing sequences for a bound to obtain the existence of positive solutions for three coupled system. To prove the uniqueness of positive solutions, we use the radial symmetry of positive solutions to transform the system into an ordinary differential system, and then integrate the system. In particular, for N = 1, we prove the uniqueness of positive solutions when 0≤β =μ1=μ2=μ3 or β >max{μ1, μ2, μ3}.